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Real subrank of order-three tensors

This paper investigates the real subrank of order-three tensors by establishing bounds relative to complex subrank, characterizing typical subranks, and providing specific constructions and results for small tensor formats and complex multiplication tensors.

Original authors: Benjamin Biaggi, Jan Draisma, Sarah Eggleston

Published 2026-08-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Benjamin Biaggi, Jan Draisma, Sarah Eggleston

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef trying to figure out the most efficient way to run a kitchen. You have a giant, complex recipe (a mathematical object called a "tensor") that takes two lists of ingredients and mixes them together to create a new dish. In the world of math, we often ask two big questions about these recipes: How many basic steps do we need to build this recipe from scratch? And, perhaps more importantly for this story, how many simple, independent "flavor bursts" can we squeeze out of this recipe if we try to use it as a machine?

Think of a "flavor burst" as a single, perfect multiplication of two numbers, like $3 \times 4 = 12$ . If your giant recipe can be tricked into doing ten of these simple multiplications at the same time, without the ingredients getting mixed up, then it has a high "subrank." This concept is crucial because it tells us the true "value" or power of a mathematical tool. If you can pack more simple multiplications into a complex system, you can solve problems faster. But here's the twist: the rules change depending on whether you are allowed to use imaginary numbers (like the square root of -1) or if you are stuck with only real, tangible numbers. This paper dives into that specific puzzle: if a machine works great in the "imaginary" world, how much of that magic can we actually capture in the "real" world?

The authors of this paper, Benjamin Biaggi, Jan Draisma, and Sarah Eggleston, are like detectives investigating these mathematical machines, specifically those that take two inputs and produce an output (order-three tensors). They wanted to know: if we know the maximum number of simple multiplications a machine can do in the complex world, what is the guaranteed minimum it can do in the real world? They proved a fascinating rule: if a machine can do $N$ complex multiplications, it can definitely do at least the square root of $N$ real multiplications. For example, if a machine is powerful enough to handle 100 complex multiplications, it is guaranteed to handle at least 10 real ones. While they initially hoped the real-world power might be closer to the complex power, they found a specific construction where the real power is indeed much smaller, confirming that the square-root rule is the best we can do in the worst-case scenario.

The team also explored the idea of "typical" behavior. In math, some shapes or machines are rare, while most are "typical." For a long time, mathematicians thought that if a machine could do 2 multiplications or 3 multiplications, it could probably do anything in between. The authors proved this is true for these real-world machines too: if 2 and 3 are possible, then 2, 3, and everything in between are all "typical" outcomes. They then went on to test specific, small-sized machines. They discovered that for a $3 \times 3 \times 5$ machine, the typical power is either 2 or 3. Even more surprisingly, they looked at a machine built from quaternions (a type of number system used in 3D graphics and physics) and found that even though it looks like it should be powerful, its real-world subrank is stuck at just 2.

Finally, the researchers looked at machines that multiply lists of complex numbers or quaternions component-by-component. They showed that for a list of $n$ complex numbers, the machine can only perform $n$ real multiplications, no matter how you try to optimize it. It's like having a device that can do $n$ complex calculations, but if you try to break it down into simple real-number steps, you can't get more than $n$ of them. They extended this finding to other number systems, proving that the "cost" of doing these multiplications in the real world is strictly limited by the size of the number system itself. This work doesn't just solve a puzzle; it sets a hard ceiling on how efficient these real-world mathematical tools can ever be.

Technical Summary: Real Subrank of Order-Three Tensors

Problem Statement
The paper investigates the subrank of real order-three tensors, a concept dual to tensor rank. While the rank of a tensor $T$ is the minimal number of rank-one tensors summing to $T$ , the subrank $Q(T)$ is the maximal integer $r$ such that the $r$ -th unit tensor $I_r$ (representing $r$ independent scalar multiplications) can be obtained from $T$ via linear slice operations in all directions.

Over algebraically closed fields (e.g., $\mathbb{C}$ ), the subrank behaves predictably: there is a unique generic subrank, and the locus of tensors with a specific subrank is well-understood. However, over the real numbers ( $\mathbb{R}$ ), the situation is more complex. Unlike the complex case, the real subrank can take multiple "typical" values (values attained on non-empty open sets in the Euclidean topology) for a given tensor format. The paper addresses the following core questions:

How does the real subrank $Q(T)$ relate to the complex subrank $Q_C(T)$ ?
What are the typical subranks for specific small tensor formats (e.g., $3 \times 3 \times 5$ , $4 \times 4 \times 4$ )?
What are the subranks of tensors associated with componentwise multiplication in real division algebras (complex numbers, quaternions, octonions)?

Methodology
The authors employ a combination of algebraic geometry, linear algebra, and topological arguments:

Duality and Bounds: The paper establishes a relationship between real and complex subranks. It utilizes the fact that the complex subrank provides an upper bound for the real subrank ( $Q(T) \leq Q_C(T)$ ) and derives a lower bound for $Q(T)$ in terms of $Q_C(T)$ .
Geometric Conditions: To bound the subrank from above, the authors use a geometric condition involving the Segre variety. Specifically, they analyze the intersection of the projectivized linear space spanned by the tensor slices with the variety of rank-one matrices. If this intersection is empty or small, the subrank is constrained.
Semialgebraic Sets and Continuity: The authors leverage the fact that sets of tensors with a given subrank are semialgebraic. They adapt arguments from Bernardi, Blekherman, and Ottaviani (BBO18) regarding tensor rank to show that if $r$ and $s$ are typical subranks with $r < s$ , then all integers between them are also typical.
Induction on Division Algebras: For the analysis of componentwise multiplication in division algebras, the paper uses an inductive argument on the dimension of the algebra and the number of components ( $n$ ), analyzing linear independence modulo off-diagonal products.

Key Contributions and Results

1. Relationship Between Real and Complex Subrank
The paper proves that for any order-three tensor $T$ , the real subrank is bounded below by the square root of the complex subrank:
$Q(T) \geq \lfloor \sqrt{Q_C(T)} \rfloor$
This bound is shown to be sharp up to a multiplicative constant via a construction communicated by Jeroen Zuiddam. Furthermore, for $n \times n \times n$ tensors where the complex subrank is maximal ( $n$ ), the authors prove a linear lower bound: $Q(T) \geq n/2$ .

2. Typical Subranks and Consecutiveness
The authors prove that typical subranks are consecutive. If $r$ and $s$ are typical subranks with $r \leq s$ , then every integer $l$ such that $r \leq l \leq s$ is also a typical subrank. This mirrors known results for tensor rank over $\mathbb{R}$ .

3. Classification of Typical Subranks for Small Formats
The paper provides a detailed classification of typical subranks for various real tensor formats $n_1 \times n_2 \times n_3$ :

$2 \times 2 \times 2$ : Typical subranks are 1 and 2.
$2 \times m \times n$ (with $m, n \geq 2$ , not both 2): The only typical subrank is 2.
$3 \times 3 \times 3$ and $3 \times 3 \times 4$ : The only typical subrank is 2.
$3 \times 3 \times 5$ : Typical subranks are 2 and 3. The authors construct a specific tensor where the subrank is 2, demonstrating that the generic subrank (3) is not the only typical value.
$3 \times 4 \times 4$ and $4 \times 4 \times 4$ : Typical subranks are 2 and 3.
- Notably, the paper shows that the quaternion multiplication tensor ( $4 \times 4 \times 4$ ) has a real subrank of 2. This is proven by showing that the linear space spanned by its slices contains no rank-one matrices (except zero), implying the subrank cannot be 3.

4. Subrank of Division Algebra Multiplication Tensors
The paper analyzes the tensor associated with componentwise multiplication in $D^n$ , where $D$ is a real division algebra of dimension $d_{\mathbb{R}}$ .

Complex Numbers ( $D=\mathbb{C}$ ): The componentwise multiplication map $f_n: \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C}^n$ has real subrank $n$ . This implies that no more than $n$ real scalar multiplications can be embedded into a device performing $n$ complex multiplications.
Quaternions ( $D=\mathbb{H}$ ): The subrank is $2n$ .
General Bound: For any real division algebra $D$ of dimension $\geq 2$ , the subrank of the componentwise multiplication map is bounded by $Q(f_n) \leq n \cdot \frac{1}{2}\dim_{\mathbb{R}}(D)$ .
Additivity: The subrank is additive for direct sums of the division algebra with itself ( $Q(f_n) = n \cdot Q(f_1)$ ).

Significance and Claims
The paper claims to resolve the question of typical subranks for several small but non-trivial tensor formats, revealing that multiple typical subranks exist even when the generic subrank is unique. This contrasts with the behavior over algebraically closed fields.

The authors emphasize that their results on division algebras are non-trivial; the subrank of tensors can be strictly superadditive, meaning the result for $n$ components does not trivially follow from the $n=1$ case. The specific finding that the quaternion multiplication tensor has subrank 2 (rather than the generic 3 for $4 \times 4 \times 4$ ) highlights the unique algebraic constraints imposed by the division algebra structure.

Finally, the paper provides a concrete upper bound for the "value" (subrank) of tensors representing complex and quaternionic multiplication, offering a limit on how many independent real scalar multiplications can be simulated by these structures. The authors conjecture that the subrank of the octonion multiplication tensor ( $8 \times 8 \times 8$ ) is 3, though they only prove the upper bound of 4.

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